# zbMATH — the first resource for mathematics

Sliding-mode pinning control of complex networks. (English) Zbl 07031757
It is considered the following model of network $\displaystyle{\dot{x}_i=f(x_i)+\sum_{j=1,j\neq i}^Nc_{ij}a_{ij}T(x_i-x_j)+u_i(t)\;,\;i=\overline{1,N}},$ where $$x_i$$ is the $$n$$-dimensional state vector of the node $$i$$; $$c_{ij}>0$$; $$T=diag\{\tau_1,\dots,\tau_n\}$$; the connection coefficients $$a_{ij}=a_{ji}$$ can be $$0$$ or $$1$$; $$a_{ii}=-k_i$$, where $\displaystyle{\sum_{j=1,j\neq i}^Na_{ij}=\sum_{j=1,j\neq i}^Na_{ji}=k_i.}$ The objective of the feedback control i.e. of the choice of $$u_i$$ is to stabilize a homogeneous stationary state $$\bar{x}$$ such that $x_1=x_2=\ldots =x_N=\bar{x}\;,\;f(\bar{x})=0.$ The choice $u_i=-c_{ii}d_iT(x_i-\bar{x})\;,\;d_i>0$ is called pinning strategy. This strategy is combined with sliding mode control. The stability of the achieved equilibrium is analyzed using a quadratic Lyapunov function.

##### MSC:
 05C82 Small world graphs, complex networks (graph-theoretic aspects) 34D06 Synchronization of solutions to ordinary differential equations 93C10 Nonlinear systems in control theory 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory